For graduate students, research engineers, practicing professionals. The primary focus of this work is to explore wavelets and filter banks over finite fields and their intersection with error control coding and data encryption. It develops the "theory of wavelet transforms over finite fields" which provides a general wavelet decomposition of sequences defined over finite fields - an approach that has a rich history in signal processing for the representation of real-valued signals, but which has been lacking in the finite field case. Along with wavelet theory on finite fields, this work introduces the first application of this theory to error correcting codes and data security. It also introduces a finite-field wavelet based public key encryption technique. - New mathematical tool (wavelet transforms over finite fields), can be used in signal processing and communications. - Novel error control coding methods based on wavelets, introduces the first application of the finite-field wavelets to error control coding. - Novel security coding methods based on wavelets, introduces the first application of the finite-field wavelets to data encryption. - Impact on basic science, brings together researchers from the areas of applied discrete mathematics, signal processing and communications research and provides new connections and new interpretations. - Impact on technology and products, provides students with orientations toward practical applications, reducing decoding complexity, and satisfying low computing requirements. - Unique treatment of coding and security, exploits finite-field wavelet techniques for coding and security and provides a state-of-the-art knowledge about the new treatment of coding and security.

Table of Contents

(NOTE: Each chapter begins with an Introduction.) 1. Introduction. 2. Background Review. Brief Review of Number Theory and Finite Fields. Discrete Fourier Transform over Finite Fields. Basefield Transforms over Finite Fields. Wavelets for Discrete-Time Signals. Previous Work on Finite-Field Wavelets. Some Basic Concepts of Error-Control Coding. Summary. 3. Finite-Field Wavelet Basis Functions. Finite-Field Discrete-Time Basis. Construction of Mother Wavelet and Scaling Function. Summary. 4. Double Circulant Wavelet Block Codes. Structure of Double Circulant Wavelet Coding. Maximum-Distance Separable Codes. Double Circulant Self-Dual Codes. Decoding Wavelet Codes. Summary. 5. Theory of Paraunitary Filter Banks Over Fields of Characteristic Two. Background Review. Unitary Matrices Over GF(2r). PU Matrices in Fields of Characteristic Two. Factorization of PU Matrices in GF(2r). Summary. 6. Arbitrary-Rate Wavelet Block Codes. Structure of Wavelet Coding. Rate 1/L Maximum-Distance Separable Codes. Arbitrary-Rate Wavelet Block Codes. Arbitrary-Rate Maximum-Distance Separable Codes. Decoding Arbitrary-Rate Wavelet Block Codes. Summary. 7. Wavelet Convolutional Codes. Structure of Wavelet Convolutional Codes. Algebraic Properties of Wavelet Convolutional Encoders. Syndrome Generators and Dual Encoders. Self-Dual and Self-Orthogonal Convolutional Codes. Time-Varying Wavelet Convolutional Codes and Bipartite Trellises. Summary. 8. Concluding Remarks. Contributions. Suggestions for Future Research. Appendix A. Proofs for Chapter 5. Appendix B. Brief Review of Trellis Structures. Bibliography. Vita.

Author Biography

Faramarz Fekri received the B.Sc. and M.Sc. degrees from Sharif University of Technology, Tehran, Iran, in 1990 and 1993, respectively. From 1995 to 1996, he was with the Telecommunication Research Laboratories (TRLabs), Calgary, Canada. There, he worked on Multicarrier Direct Sequence Spread Spectrum for Mobile Communication systems and was a recipient of the TRLabs' distinction award. In Fall of 1996, he started his Ph.D. studies at Georgia Tech. and received the Ph.D. degree in 2000. Since 2000, Dr. Fekri has been with the faculty of the School of ECE at the Georgia Institute of Technology, where he is now a full professor. He is a member of the Georgia Tech Broadband Institute (GTBI), the Center for Signal and Image Processing (CSIP), and the Georgia Tech Information Security Center (GTISC). Farshid Delgosha received the Ph.D. degree in Electrical and Computer Engineering from Georgia Institute of Technology, Atlanta, GA, USA, in 2007. The focus of his Ph.D. dissertation was in wavelet transform over finite fields, algebraic cryptography, and security aspects of wireless sensor networks. In 2006, he received the outstanding research award from the Center for Signal and Image Processing (CSIP), School of ECE, Georgia Institute of Technology. Since 2007, he has been with the faculty of Electrical and Computer Engineering Department at the New York Institute of Technology. He has been an IEEE member since 2002.